Free probability, Planar algebras, Subfactors and Random Matrices

Details Free probability, Planar algebras, Subfactors and Random Matrices Free probability, Planar algebras, Subfactors and Random Matrices

2010-09-05

arxiv.org/abs/1009.0939v1

Proceedings of the International Congress of Mathematicians (Hyderabad, India, 2010), Vol. III, Hindustan Book Agency, 2010, p

Mathematics

Operator Algebras

CM 2010 Proceedings text

Scientific paper

To a planar algebra P in the sense of Jones we associate a natural non- commutative ring, which can be viewed as the ring of non-commutative polynomials in several indeterminates, invariant under a symmetry encoded by P. We show that this ring carries a natural structure of a non-commutative probability space. Non-commutative laws on this space turn out to describe random matrix ensembles possessing special sym- metries. As application, we give a canonical construction of a subfactor and its symmetric enveloping algebra associated to a given planar algebra P. This talk is based on joint work with A. Guionnet and V. Jones.

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